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© Boardworks Ltd 2004 1 of 55 D2D2 D2D2 D2D2 D2D2 D2D2 D4.1 The language of probability Contents D4 Probability D4.5 Experimental probability D4.2 The probability scale D4.4 Probability diagrams D4.3 Calculating probability

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© Boardworks Ltd 2004 2 of 55 The language of probability Probability is a measurement of the chance or likelihood of an event happening. Describe the chance of drawing a red marble. Unlikely غير مرجح Likely مرجح Certain مؤكد Impossible مستحيل even Chance متساوي الفرصة

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© Boardworks Ltd 2004 3 of 55 The probability scale The chance of an event happening can be shown on a probability scale. impossiblecertaineven chanceunlikely likely Less likelyMore likely Meeting with King Henry VIII A day of the week starting with a T The next baby born being a boy Getting a number > 2 when roll a fair dice A square having four right angles

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© Boardworks Ltd 2004 4 of 55 Fair games A game is played with marbles in a bag. One of the following bags is chosen for the game. The teacher then pulls a marble at random from the chosen bag: If a red marble is pulled out of the bag, the girls get a point. If a blue marble is pulled out of the bag, the boys get a point. Which would be the fair bag to use? bag a bag c bag b

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© Boardworks Ltd 2004 5 of 55 Fair games A game is fair if all the players have an equal chance of winning. Which of the following games are fair? A dice is thrown. If it lands on a prime number team A gets a point, if it doesn’t team B gets a point. There are three prime numbers (2, 3 and 5) and three non-prime numbers (1, 4 and 6). Yes, this game is fair.

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© Boardworks Ltd 2004 6 of 55 Fair games Nine cards numbered 1 to 9 are used and a card is drawn at random. If a multiple of 3 is drawn team A gets a point. If a square number is drawn team B gets a point. If any other number is drawn team C gets a point. There are three multiples of 3 (3, 6 and 9). No, this game is not fair. Team C is more likely to win. There are three square numbers (1, 4 and 9). There are four numbers that are neither square nor multiples of 3 (2, 5, 7 and 8).

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© Boardworks Ltd 2004 7 of 55 Fair games A spinner has five equal sectors numbered 1 to 5. The spinner is spun many times. If the spinner stops on an even number team A gets 3 points. If the spinner stops on an odd number team B gets 2 points. 1 2 3 4 5 Suppose the spinner is spun 50 times. We would expect the spinner to stop on an even number 20 times and on an odd number 30 times. Team A would score 20 × 3 points = 60 points Team B would score 30 × 2 points = 60 points Yes, this game is fair.

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© Boardworks Ltd 2004 8 of 55 Bags of counters You are only allowed to choose one counter at random from one of the bags. Which of the bags is most likely to win a prize? Choose a blue counter and win a prize! bag a bag b bag c

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© Boardworks Ltd 2004 9 of 55 The probability scale The chance of an event happening can be shown on a probability scale. impossiblecertaineven chanceunlikely likely Less likelyMore likely Meeting with King Henry VIII A day of the week starting with a T The next baby born being a boy Getting a number > 2 when roll a fair dice A square having four right angles

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© Boardworks Ltd 2004 10 of 55 The probability scale We measure probability on a scale from 0 to 1. If an event is impossible or has no probability of occurring then it has a probability of 0. If an event is certain it has a probability of 1. This can be shown on the probability scale as: Probabilities are written as fractions, decimal and, less often, as percentages between 0 and 1. 0½1 impossiblecertaineven chance

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© Boardworks Ltd 2004 11 of 55 The probability scale

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© Boardworks Ltd 2004 12 of 55 D2D2 D2D2 D2D2 D2D2 D2D2 D4.3 Calculating probability Contents D4 Probability D4.1 The language of probability D4.5 Experimental probability D4.2 The probability scale D4.4 Probability diagrams

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© Boardworks Ltd 2004 13 of 55 Higher or lower

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© Boardworks Ltd 2004 14 of 55 Listing possible outcomes When you roll a fair dice you are equally likely to get one of six possible outcomes: 1 6 1 6 1 6 1 6 1 6 1 6 Since each number on the dice is equally likely the probability of getting any one of the numbers is 1 divided by 6 or. 1 6

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© Boardworks Ltd 2004 15 of 55 Calculating probability What is the probability of the following events? P(tails) = 1 2 P(red) = 1 4 P(7 of ) = 1 52 P(Friday) = 1 7 2) This spinner stopping on the red section? 3) Drawing a seven of hearts from a pack of 52 cards? 4) A baby being born on a Friday? 1) A coin landing tails up?

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© Boardworks Ltd 2004 16 of 55 Calculating probability If the outcomes of an event are equally likely then we can calculate the probability using the formula: Probability of an event = Number of successful outcomes Total number of possible outcomes For example, a bag contains 1 yellow, 3 green, 4 blue and 2 red marbles. What is the probability of pulling a green marble from the bag without looking? P(green) = 3 10 or 0.3 or 30%

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© Boardworks Ltd 2004 17 of 55 Calculating probability This spinner has 8 equal divisions: a)a red sector? b)a blue sector? c)a green sector? What is the probability of the spinner landing on a) P(red) = 2 8 = 1 4 b) P(blue) = 1 8 c) P(green) = 4 8 = 1 2

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© Boardworks Ltd 2004 18 of 55 Calculating probability A fair dice is thrown. What is the probability of getting a)a 2? b)a multiple of 3? c)an odd number? d)a prime number? e)a number bigger than 6? f)an integer? a) P(2) = 1 6 b) P(a multiple of 3) = 2 6 = 1 3 c) P(an odd number) = 3 6 = 1 2

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© Boardworks Ltd 2004 19 of 55 Calculating probability A fair dice is thrown. What is the probability of getting a)a 2? b)a multiple of 3? c)an odd number? d)a prime number? e)a number bigger than 6? f)an integer? d) P(a prime number) = 3 6 e) P(a number bigger than 6) = f) P(an integer) = 6 6 = 1 = 1 2 0 Don’t write 0 6

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© Boardworks Ltd 2004 20 of 55 Calculating probabilities Answer these questions giving each answer as a fraction or 0 or 1.

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© Boardworks Ltd 2004 21 of 55 The probability of an event not occurring If the probability of an event occurring is p then the probability of it not occurring is 1 – p. The following spinner is spun once: What is the probability of it landing on the yellow sector? P(yellow) = 1 4 What is the probability of it not landing on the yellow sector? P(not yellow) = 3 4

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© Boardworks Ltd 2004 22 of 55 The probability of an event not occurring The probability of a factory component being faulty is 0.03. What is the probability of a randomly chosen component not being faulty? P(not faulty) = 1 – 0.03 =0.97 The probability of pulling a picture card out of a full deck of cards is. What is the probability of not pulling out a picture card? 3 13 P(not a picture card) = 1 – = 3 13 10 13

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© Boardworks Ltd 2004 23 of 55 The probability of an event not occurring The following table shows the probabilities of 4 events. For each one work out the probability of the event not occurring. Event Probability of the event occurring Probability of the event not occurring ABCD 3 5 9 20 0.77 8% 2 5 11 20 0.23 92%

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© Boardworks Ltd 2004 24 of 55 The probability of an event not occurring There are 60 sweets in a bag. What is the probability that a sweet chosen at random from the bag is: a) Not a cola bottle 5 6 P(not a cola bottle) = b) Not a teddy 45 60 P(not a teddy) = 10 are cola bottles, 1 4 are fried eggs,the rest are teddies.20 are hearts, = 3 4

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© Boardworks Ltd 2004 25 of 55 Adding mutually exclusive outcomes If two outcomes are mutually exclusive then their probabilities can be added together to find their combined probability. What is the probability that a card is a moon or a sun? P(moon) = 1 3 andP(sun) = 1 3 Drawing a moon and drawing a sun are mutually exclusive outcomes so, P(moon or sun) = P(moon) + P(sun) = 1 3 + 1 3 = 2 3 For example, a game is played with the following cards:

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© Boardworks Ltd 2004 26 of 55 Adding mutually exclusive outcomes What is the probability that a card is yellow or a star? P(yellow card) = 1 3 andP(star) = 1 3 Drawing a yellow card and drawing a star are not mutually exclusive outcomes because a card could be yellow and a star. P (yellow card or star) cannot be found simply by adding. P(yellow card or star) = We have to subtract the probability of getting a yellow star. 1 3 + 1 3 – 1 9 = 3 + 3 – 1 9 = 5 9

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© Boardworks Ltd 2004 27 of 55 The sum of all mutually exclusive outcomes The sum of all mutually exclusive outcomes is 1. For example, a bag contains red counters, blue counters, yellow counters and green counters. P(blue) = 0.15P(yellow) = 0.4P(green) = 0.35 What is the probability of drawing a red counter from the bag? P(blue or yellow or green) = 0.15 + 0.4 + 0.35 =0.9 P(red) = 1 – 0.9 =0.1

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© Boardworks Ltd 2004 28 of 55 Finding all possible outcomes of two events Two coins are thrown. What is the probability of getting two heads? Before we can work out the probability of getting two heads we need to work out the total number of equally likely outcomes. There are three ways to do this: 1) We can list them systematically. Using H for heads and T for tails, the possible outcomes are: TH and HT are separate equally likely outcomes. HH.TT,TH,HT,

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© Boardworks Ltd 2004 29 of 55 Finding all possible outcomes of two events 2) We can use a two-way table. Second coin HT H T First coin HHHT THTT From the table we see that there are four possible outcomes one of which is two heads so, P(HH) = 1 4

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© Boardworks Ltd 2004 30 of 55 Finding all possible outcomes of two events 3) We can use a probability tree diagram. First coin H T Second coin H T H T Outcomes HH HT TH TT Again we see that there are four possible outcomes so, P(HH) = 1 4

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© Boardworks Ltd 2004 31 of 55 Finding the sample space A red dice and a blue dice are thrown and their scores are added together. What is the probability of getting a total of 8 from both dice? There are several ways to get a total of 8 by adding the scores from two dice. We could get a 2 and a 6,a 3 and a 5,a 4 and a 4, a 5 and a 3,or a 6 and a 2. To find the set of all possible outcomes, the sample space, we can use a two-way table.

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© Boardworks Ltd 2004 32 of 55 Finding the sample space + 234567 345678 456789 5678910 6789 11 789101112 From the sample space we can see that there are 36 possible outcomes when two dice are thrown. Five of these have a total of 8. 888 88 P(8) = 5 36

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© Boardworks Ltd 2004 33 of 55 D2D2 D2D2 D2D2 D2D2 D2D2 D4.5 Experimental probability Contents D4 Probability D4.1 The language of probability D4.2 The probability scale D4.4 Probability diagrams D4.3 Calculating probability

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© Boardworks Ltd 2004 34 of 55 Estimating probabilities based on data Suppose 1000 people were asked whether they were left- or right-handed. Of the 1000 people asked 87 said that they were left- handed. If we repeated the survey with a different sample the results would probably be slightly different. From this we can estimate the probability of someone being left-handed as or 0.087. 87 1000 The more people we asked, however, the more accurate our estimate of the probability would be.

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© Boardworks Ltd 2004 35 of 55 Relative frequency The probability of an event based on data from an experiment or survey is called the relative frequency. Relative frequency is calculated using the formula: Relative frequency = Number of successful trials Total number of trials For example, Ben wants to estimate the probability that a piece of toast will land butter-side-down. He drops a piece of toast 100 times and observes that it lands butter-side-down 65 times. Relative frequency = 65 100 = 13 20

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© Boardworks Ltd 2004 36 of 55 Relative frequency Sita wants to know if her dice is fair. She throws it 200 times and records her results in a table: NumberFrequencyRelative frequency 131 227 338 430 542 632 Is the dice fair? 31 200 27 200 38 200 30 200 42 200 32 200 = 0.155 = 0.135 = 0.190 = 0.150 = 0.210 = 0.160

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© Boardworks Ltd 2004 37 of 55 Experimental probability

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© Boardworks Ltd 2004 38 of 55 Expected frequency The theoretical probability of an event is its calculated probability based on equally likely outcomes. Expected frequency = theoretical probability × number of trials If you rolled a dice 300 times, how many times would you expect to get a 5? The theoretical probability of getting a 5 is. 1 6 So, expected frequency = × 300 = 1 6 50 If the theoretical probability of an event can be calculated, then when we do an experiment we can work out the expected frequency.

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© Boardworks Ltd 2004 39 of 55 Expected frequency If you tossed a coin 250 times how many times would you expect to get a tail? Expected frequency = × 250 = 1 2 125 If you rolled a fair dice 150 times how many times would you expect to a number greater than 2? Expected frequency = × 150 = 2 3 100

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© Boardworks Ltd 2004 40 of 55 Spinners experiment

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* Write the sample space ( all possible results ) when rolling a fair dice Worksheet ( 1 ) Zayed althani school Math department Mohamad badawi : hamadaa_math@yahoo.com اكتب فضاء العينة ( مجموعة جميع النواتج الممكنة ) عند إلقاء حجر نرد منتظم Coin 2 HT Coin 1 H T 1) Complete the table to show all possible results. Use the words : impossible, unlikely, even chance, likely, certain to describe the following events : 1) The upper face is a number greater than 5. ………………. 2) The upper face is a prime number. ………………. * You toss 2 coins together 1) You will get 2 heads ………………. 2) At least one head ………………. Use the words : impossible, unlikely, even chance, likely, certain to describe the following events : 3) You will get one tail exactly………………. استخدم المصطلحات : مستحيل ، غير مرجح ، متساوي الفرصة ، مرجح ، مؤكد لتصف الأحداث التالية : ظهور عدد اكبر من 5 على الوجه العلوي ظهور عدد أولي على الوجه العلوي ألقيت قطعتي نقود معاً ستحصل على صورتين ستظهر صورة واحدة على الأقل ستظهر الكتابة مرة واحدة بالضبط أكمل الجدول لتبين جميع النواتج الممكنة

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2 cards were randomly drawn from a deck of 52 cards Worksheet ( 2 ) Zayed althani school Math department Mohamad badawi : hamadaa_math@yahoo.com تم سحب ورقتين من علبة لعب الورق ( الشدة ) التي تحوي 52 ورقة Card 2 Card 1 : Spade بستوني : Clubs سباتي : Diamond ديناري : Heart كبه ( قلب ) 1) Complete the table to show all possible results. Use the words : impossible, unlikely, even chance, likely, certain to describe the following events : 1) The 2 cards are of the same color. ………………. 2) The 2 cards are spades. ………………. 3) One of the cards was green. ………………. 4) The 2 cards are either red or black or red and black………………. 5) At least one of the 2 cards wasn’t a picture. ………………. أكمل الجدول لتبين جميع النواتج الممكنة استخدم المصطلحات : مستحيل ، غير مرجح ، متساوي الفرصة ، مرجح ، مؤكد لتصف الأحداث التالية : البطاقتان من نفس اللون البطاقتان من نوع البستوني احد البطاقتين خضراء البطافتان إما حمراوتان أو سوداوتان أو حمراء وسوداء على الأقل احدهما ليست صورة

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